Numerical modeling, weather predictability and forecast

Seminar X-ENS-UPS – 12 May 2011Philippe Drobinski 1/27

Numerical modeling, weather predictability and forecast

Philippe Drobinski

Laboratoire de Météorologie Dynamique

Ecole Polytechnique

[email protected]


Outline

� Elements of numerical modelingModeling turbulent flow

Subgrid scale parameterization

Initial and boundary conditions

� Theories on atmospheric predictabilityLorenz attractor

Initial condition problem

Sensitivity to parameterization

� Weather forecast and climate predictionA historical perspective

Deterministic and ensemble forecasts

Weather forecast and climate projections


∆=100 km

Large-scale numerical weather prediction

(NWP) models

∆=10 km

Meso-scale numerical weather prediction

(NWP) models

∆<2 km

Large-eddy simulation

(LES) models

Elements of numerical modelingTheories on atmospheric precitabilityWeather forecast and climate prediction

Which scales are resolved?

Modeling turbulent Subgrid scale parameterizationInitial and boundary conditions


'uu~u iii +=


Separation into large and small scales


Reynolds averaged equations: ξ = ξ + ξ’

j

ij

j3ij

ij

ij

i

xuf

x

p

x

uu

t

u

∂τ∂

−ε+∂∂

−=∂∂+

∂∂

j

j

j

*

j

V

pj

jx

q

x

QEL

C

1

xu

t ∂∂

−

∂∂

+ρ

−=∂

θ∂+∂θ∂

'u'u jiij =τ

''uq jj θ=

SGS stress

SGS flux

(SGS: sub-grid scale)


Limited-area models (LAM)General circulation models (GCM)

Types of models


Discretization of the conservation equations (partial differential equations)

Finite difference method Finite element methodFinite volume methodBoundary element method

Finite element method suited for complex geometryFinite difference method easy to implement

More details are out of the scope of this course





15103rd order

1062nd order

631st order

Number of unknowns

Number of equations

EquationMomentPrognostic equation for:

j'j

'ii x/uut/U ∂∂−=∂∂ L

k'k

'j

'i

'j

'i x/uuut/uu ∂∂−=∂∂ L

m'm

'k

'j

'i

'k

'j

'i x/uuuut/uuu ∂∂−=∂∂ L

iU

'j

'iuu

'k

'j

'i uuu

Need for closure equations referred as parameterization




Data assimilation principle:Estimate the atmospheric state of day D-1 (first guess)Perform a simulation from day D-1 yesterday to day DQuantify model/measurement differenceModify the first guess in order to decrease the differenceRe-iterate while necessary

Reconstruct 3D meteorological information from incomplete dataset � data assimilation

Data assimilation

3D state of the atmosphere is needed to provide initial and boundary conditions to the NWP


Numerical approximation of convection equation

Boussinesq equations

Lorenz attractorInitial condition problemSensitivity to parameterization

( )[ ]

Θ

∂∂+

∂∂κ=Θ

∂∂+

∂∂ν+−α−−

∂∂

ρ−=

∂∂+

∂∂ν+

∂∂

ρ−=

=∂∂+

∂∂

2

2

2

2

2

2

2

2

0

0

2

2

2

2

0

zxdt

d

wzx

TT1gz

p1

dt

dw

uzxx

p1

dt

du

0z

w

x

u

∂∂−

∂∂

∂∂+

∂∂ν+

∂Θ∂α=

∂∂−

∂∂

z

u

x

w

zxxg

z

u

x

w

dt

d2

2

2

2

( )[ ]00 TT1 −α−ρ=ρ


Boussinesq equations with


ψ

∂∂+

∂∂σ+

∂θ∂σ=ψ

∂∂+

∂∂

θ

∂∂+

∂∂+

∂ψ∂=θ

2

2

2

2

2

2

2

2

2

2

2

2

2

zxxzxdt

d

zxxR

dt

d


( ) ( )L

Ha;numberRayleigh

HgR;numberandtlPr

3

=κν

∆Τα=

κν=σ

Non-dimensional equations

Solutions in a periodic box of size 2L×2H (Saltzman, 1962)

with

( ) ( ) ( )

( ) ( ) ( )∑

∑+π

+π

θ=θ

ψ=ψ

n,m

nzamxi

n,m

n,m

nzamxi

n,m

etˆt,z,x

etˆt,z,x



bZXYt

Z

YrXXZt

Y

YXt

X

−=∂∂

−+−=∂∂

σ+σ−=∂∂


Lorenz model (Lorenz, 1963; Saltzman 1962 simplification)

( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )z2sintZzsinaxcos2tYt,z,xR

zsinaxsin2tXt,z,x1a

a

1

c

2

π−ππ=θπ

ππ=ψ+−

Equilibrium states

1a

4b;

R

Rr;

2

c +==

κν=σwith

Perturbation equations

( ) 1rZ;1rbYX

0ZYX

−=−±==

===

( )

−−−−

σσ−=

∂∂

0

0

0

0

0

0

z

y

x

bXY

X1Zr

0

z

y

x

t

a problem with 3 degrees of freedom

Unstable for σ=10,b=8/3 and r=28





Ensemble forecast on Lorenz attractor

Example of how forecast uncertainty can vary depending on the location of the initial state

http://www.wmo.ch/pages/prog/www/DPS/TC-DPFS-2002/Papers-Posters/Keynote-Richardson.pdf


( )

( ) 2

2

2

6

222

1

6

111

q,Jt

q

q,Jt

q

ψ∇κ−ψ∇ν−=ψ+∂

∂

ψ∇ν−=ψ+∂

∂

Quasi-geostrophic model

Haidvogel and Held (1980)



x

y

« First » numerical weather prediction model

Anticyclones

Cyclones



Initial error growth – Lyapunov exponent – predictability time

( )qft

q=

∂∂ ( )

( ) qq0tq

q0tq

0

0ref

δ+====

2 simulations: qref, q

Omrani et al. (2011)

Lyapunov exponent λPredictability time τP ~ 1/λ

( ) ( ) t

ref e~tqtq λ−

( ) ( ) ?tqtq ref =−

qref(t) q(t)≠≠≠≠t>>τp



Measured fluxes

Eddy diffusion

Non linear

Mixed

Evaluation of closure model sensitivity in off-line mode

( )

k

j

k

i2

nl

ij

2

S

mix

ij

x

u~

x

u~C

S~

S~

C2

∂∂

∂∂∆+

∆−=τ

Non linear models

Non-linear models are unstable (simulations blow up) when applied aloneThey are used in linear combination with eddy-diffusion model

k

j

k

i2

nl

nl

ijx

u~

x

u~C

∂∂

∂∂∆=τ

Mixed models



( ) ij

2

sij S~

S~

C2 ∆−=τEddy-diffusivity model


Bjerknes, V. (1904). Das Problem der Wettervorhersage, betrachtet vom Standpunkte der Mechanik und der Physik. Meteorologische Zeitschrift, 21, 1-7 Vilhelm Bjerknes

1862-1951

If, as every scientifically inclined individual believes, atmospheric conditions develop according to natural laws from their precursors, it follows that the necessary and sufficient conditions for a rational solution of the problems of meteorological prediction are:

the condition of the atmosphere must be known at a specific time with sufficient accuracythe laws must be known, with sufficient accuracy, which determine the development of one weather condition from another.

Lewis Fry Richardson1881-1953

First numerical integration … by hand in 1922 by L.W. Richardson � complete failure. Richardson estimates that a real-time weather forecast needs 30000 people making calculation simultaneously


A historical perspectiveDeterministic and ensemble forecastsWeather forecast and climate projections


Hans Ertel1904-1971

H. Ertelfinds out the potential vorticity and its conservation.develops the quasi-geostrophic theory of mid-latitude atmospheric dynamics.

J.G. Charney, R. Fjørtoft, and J. von Neumann, 1950: Numerical integration of the barotropic vorticity equation. Tellus, 2, 237–254

First « successfull » numerical weather forcast with « general circulation model » in 1950 at ENIAC (J. Charney, P. Thompson, L. Gates, R. Fjörtoft).



Jule Gregory Charney1917-1981

Ertel, H., 1942: Ein neuer hydrodynamischerErhaltungssatz. - Meteorol. Z. 59, 277–281


Group photo of the 1st International Symposium on Numerical Weather Prediction held in Tokyo on 78-13 November 1960 (Syono, 1962)




Two similar initial states can lead to different forecasts !

(Lorenz, 1982)

Growth of forecast error for operational 10-day forecast at ECMWF avearged over a 100-day period). Global root-mean-square 500-hPa height difference (m) between j-day and k-day forecast.



Deterministic forecast


Importance of observations (inaccurate and unevenly spaced)




Visual analysis of initial conditions

Production of new initial conditions

Example of Klaus windstorm (January, 23rd, 2009)



Importance of the forecasters


Example of Klaus windstorm (January,

23rd, 2009)



Impact on meteorological forecast


(Shapiro and Thorpe, 2004)




Ensemble forecast



Distribution of initial condition uncertainty

Deterministic forecast

Real state of the atmosphere

Forecast of the uncertainty

Principle




Methods

Multi-analyses (analyses from different weather services, e.g. Météo-France, ECMWF, NCEP; perturbed initial conditions)

Multi-models

Need for a synthetic probabilitic representation





Climate = statistical distribution of timeA climate model is a tool for scientific investigation for:

• understanding of the past and current climates on Earth, Mars, Venus ...• investigating the causes of its variations (forcings / internal variability) • producing projections for the future

• Providing an accurate forecast up to 10 days (typical predictability time) does not mean we can not produce reliable projections for the future, which are statistical representation of the atmospheric and/or oceanic states




Next time…

Thank you for your attentionThank you for your attention………… any questions?any questions?

Documents

Numerical modeling, weather predictability and forecast